# Analysis and Design of Robust Max Consensus for Wireless Sensor Networks

## TL;DR

This paper introduces a robust distributed max consensus algorithm for wireless sensor networks that accounts for communication noise, using max-plus algebra and ergodic theory to analyze and bound the estimation error.

## Contribution

It develops a novel noise-robust max consensus algorithm and provides theoretical bounds on its performance in noisy network conditions.

## Key findings

- The growth rate of node estimates is bounded by network and noise parameters.
- The proposed algorithm maintains accurate max estimates despite additive noise.
- Simulation results confirm the theoretical bounds and robustness of the method.

## Abstract

A novel distributed algorithm for estimating the maximum of the node initial state values in a network, in the presence of additive communication noise is proposed. Conventionally, the maximum is estimated locally at each node by updating the node state value with the largest received measurements in every iteration. However, due to the additive channel noise, the estimate of the maximum at each node drifts at each iteration and this results in nodes diverging from the true max value. Max-plus algebra is used as a tool to study this ergodic process. The subadditive ergodic theorem is invoked to establish a constant growth rate for the state values due to noise, which is studied by analyzing the max-plus Lyapunov exponent of the product of noise matrices in a max-plus semiring. The growth rate of the state values is upper bounded by a constant which depends on the spectral radius of the network and the noise variance. Upper and lower bounds are derived for both fixed and random graphs. Finally, a two-run algorithm robust to additive noise in the network is proposed and its variance is analyzed using concentration inequalities. Simulation results supporting the theory are also presented.